B is a matrix (3 2)and C is the matrix (9 -3)
(2 2). (2 1)
Of a matrix is a 2×2 matrix and A×B=C determine matrix of A
To determine the matrix A, we can use the equation A x B = C and solve for A.
Let's represent the elements of matrix A as follows:
A = | a b |
| c d |
Using matrix multiplication, the equation A x B = C becomes:
| 3 2 | | a b | = | 9 -3 |
| c d | | 2 1 |
Multiplying the matrices on the left side, we have:
| 3a + 2c 3b + 2d | = | 9 -3 |
| 2 1 |
From this equation, we can equate the corresponding elements:
3a + 2c = 9,
3b + 2d = -3,
a + c = 2,
b + d = 1.
From the first equation, we can solve for a:
3a = 9 - 2c,
a = (9 - 2c)/3.
Similarly, from the third equation, we can solve for c:
c = 2 - a.
Substituting the value of c into the first equation, we have:
3a = 9 - 2(2 - a),
3a = 9 - 4 + 2a,
a - 2a = 5,
-a = 5,
a = -5.
Substituting the value of a back into c = 2 - a:
c = 2 - (-5),
c = 7.
Now that we have found the values of a and c, we can solve for b and d using the second and fourth equations:
3b + 2d = -3,
b + d = 1.
From the fourth equation, we have:
b = 1 - d.
Substituting this value of b into the second equation, we have:
3(1 - d) + 2d = -3,
3 - 3d + 2d = -3,
-d = -6,
d = 6.
substituting the value of d back into b = 1 - d:
b = 1 - 6,
b = -5.
Therefore, the matrix A is:
A = | -5 -5 |
| 7 6 |
To determine the matrix A, we need to find the inverse of matrix B.
Given that B is a matrix (3 2) and (2 2), we can represent it as:
B = |3 2|
|2 2|
To find the inverse of B, we can use the formula:
A^(-1) = (1 / det(B)) * adj(B)
where det(B) is the determinant of B and adj(B) is the adjugate of B.
First, let's calculate det(B):
det(B) = (3 * 2) - (2 * 2)
= 6 - 4
= 2
Next, we need to find the adjugate of B. The adjugate of B is obtained by swapping the elements on the main diagonal and changing the sign of the elements on the other diagonal:
adj(B) = | 2 -2|
|-2 3|
Now, let's find A^(-1) using the formula mentioned earlier:
A^(-1) = (1 / det(B)) * adj(B)
= (1 / 2) * | 2 -2|
|-2 3|
Multiplying 1/2 with the adjugate matrix:
A^(-1) = | 1 -1|
|-1 3 |
Therefore, the matrix A is:
A = | 1 -1 |
|-1 3 |